Theorem elmpps 33248

Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mppsval.p	⊢ 𝑃 = (mPreSt‘𝑇)
mppsval.j	⊢ 𝐽 = (mPPSt‘𝑇)
mppsval.c	⊢ 𝐶 = (mCls‘𝑇)

Assertion

Ref	Expression
elmpps	⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))

Proof of Theorem elmpps

Dummy variables 𝑎 𝑑 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ot 4550	. . 3 ⊢ ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
2		mppsval.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
3		mppsval.j	. . . 4 ⊢ 𝐽 = (mPPSt‘𝑇)
4		mppsval.c	. . . 4 ⊢ 𝐶 = (mCls‘𝑇)
5	2, 3, 4	mppsval 33247	. . 3 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}
6	1, 5	eleq12i 2830	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
7		oprabss 7317	. . . 4 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ ((V × V) × V)
8	7	sseli 3896	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
9	2	mpstssv 33214	. . . . . 6 ⊢ 𝑃 ⊆ ((V × V) × V)
10	9	sseli 3896	. . . . 5 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
11	1, 10	eqeltrrid 2843	. . . 4 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
12	11	adantr 484	. . 3 ⊢ ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)) → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
13		opelxp 5587	. . . 4 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
14		opelxp 5587	. . . . 5 ⊢ (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
15		simp1 1138	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → 𝑑 = 𝐷)
16		simp2 1139	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ℎ = 𝐻)
17		simp3 1140	. . . . . . . . . 10 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴)
18	15, 16, 17	oteq123d 4799	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ⟨𝑑, ℎ, 𝑎⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
19	18	eleq1d 2822	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ↔ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃))
20	15, 16	oveq12d 7231	. . . . . . . . 9 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (𝑑𝐶ℎ) = (𝐷𝐶𝐻))
21	17, 20	eleq12d 2832	. . . . . . . 8 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → (𝑎 ∈ (𝑑𝐶ℎ) ↔ 𝐴 ∈ (𝐷𝐶𝐻)))
22	19, 21	anbi12d 634	. . . . . . 7 ⊢ ((𝑑 = 𝐷 ∧ ℎ = 𝐻 ∧ 𝑎 = 𝐴) → ((⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)) ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
23	22	eloprabga 7318	. . . . . 6 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
24	23	3expa 1120	. . . . 5 ⊢ (((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
25	14, 24	sylanb 584	. . . 4 ⊢ ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
26	13, 25	sylbi 220	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻))))
27	8, 12, 26	pm5.21nii 383	. 2 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))
28	6, 27	bitri 278	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝐴 ∈ (𝐷𝐶𝐻)))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2110  Vcvv 3408  ⟨cop 4547  ⟨cotp 4549   × cxp 5549  ‘cfv 6380  (class class class)co 7213  {coprab 7214  mPreStcmpst 33148  mClscmcls 33152  mPPStcmpps 33153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-ot 4550  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpst 33168  df-mpps 33173

This theorem is referenced by:  mthmpps  33257  mclspps  33259